Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userMinimization of divergences on sets of signed measures
245
0
0.0
(
0
)
Authors
Michel Broniatowski
Ask ChatGPT about the research
No Arabic abstract
We consider the minimization problem of $phi$-divergences between a given probability measure $P$ and subsets $Omega$ of the vector space $mathcal{M}_mathcal{F}$ of all signed finite measures which integrate a given class $mathcal{F}$ of bounded or unbounded measurable functions. The vector space $mathcal{M}_mathcal{F}$ is endowed with the weak topology induced by the class $mathcal{F}cup mathcal{B}_b$ where $mathcal{B}_b$ is the class of all bounded measurable functions. We treat the problems of existence and characterization of the $phi$-projections of $P$ on $Omega$. We consider also the dual equality and the dual attainment problems when $Omega$ is defined by linear constraints.
rate research
Read More
The divergence minimization problem plays an important role in various fields. In this note, we focus on differentiable and strictly convex divergences. For some minimization problems, we show the minimizer conditions and the uniqueness of the minimizer without assuming a specific form of divergences. Furthermore, we show geometric properties related to the minimization problems.
This paper establishes It^os formula along a flow of probability measures associated with gene-ral semimartingales. This generalizes existing results for flow of measures on It^o processes. Our approach is to first prove It^os formula for cylindrical polynomials and then use function approximation and localization techniques for the general case. This general form of It^os formula enables derivation of dynamic programming equations and verification theorems for McKean-Vlasov controls with jump diffusions and for McKean-Vlasov mixed regular-singular control problems. It also allows for generalizing the classical relation between the maximum principle and the dynamic programming principle to the McKean-Vlasov singular control setting, where the adjoint process is expressed in term of the derivative of the value function with respect to probability measures.
We consider a large class of random geometric graphs constructed from samples $mathcal{X}_n = {X_1,X_2,ldots,X_n}$ of independent, identically distributed observations of an underlying probability measure $ u$ on a bounded domain $Dsubset mathbb{R}^d$. The popular `modularity clustering method specifies a partition $mathcal{U}_n$ of the set $mathcal{X}_n$ as the solution of an optimization problem. In this paper, under conditions on $ u$ and $D$, we derive scaling limits of the modularity clustering on random geometric graphs. Among other results, we show a geometric form of consistency: When the number of clusters is a priori bounded above, the discrete optimal partitions $mathcal{U}_n$ converge in a certain sense to a continuum partition $mathcal{U}$ of the underlying domain $D$, characterized as the solution of a type of Kelvins shape optimization problem.
We consider Gibbs distributions on permutations of a locally finite infinite set $Xsubsetmathbb{R}$, where a permutation $sigma$ of $X$ is assigned (formal) energy $sum_{xin X}V(sigma(x)-x)$. This is motivated by Feynmans path representation of the quantum Bose gas; the choice $X:=mathbb{Z}$ and $V(x):=alpha x^2$ is of principal interest. Under suitable regularity conditions on the set $X$ and the potential $V$, we establish existence and a full classification of the infinite-volume Gibbs measures for this problem, including a result on the number of infinite cycles of typical permutations. Unlike earlier results, our conclusions are not limited to small densities and/or high temperatures.
Given a target function $U$ to minimize on a finite state space $mathcal{X}$, a proposal chain with generator $Q$ and a cooling schedule $T(t)$ that depends on time $t$, in this paper we study two types of simulated annealing (SA) algorithms with generators $M_{1,t}(Q,U,T(t))$ and $M_{2,t}(Q,U,T(t))$ respectively. While $M_{1,t}$ is the classical SA algorithm, we introduce a simple and improved variant that we call $M_{2,t}$ which provably converges faster. When $T(t) > c_{M_2}/log(t+1)$ follows the logarithmic cooling schedule, our proposed algorithm is strongly ergodic both in total variation and in relative entropy, and converges to the set of global minima, where $c_{M_2}$ is a constant that we explicitly identify. If $c_{M_1}$ is the optimal hill-climbing constant that appears in logarithmic cooling of $M_{1,t}$, we show that $c_{M_1} geq c_{M_2}$ and give simple conditions under which $c_{M_1} > c_{M_2}$. Our proposed $M_{2,t}$ thus converges under a faster logarithmic cooling in this regime. The other situation that we investigate corresponds to $c_{M_1} > c_{M_2} = 0$, where we give a class of fast and non-logarithmic cooling schedule that works for $M_{2,t}$ (but not for $M_{1,t}$). In addition to these asymptotic convergence results, we compare and analyze finite-time behaviour between these two annealing algorithms as well. Finally, we present two algorithms to simulate $M_{2,t}$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
